Question: The measure of angle $ACB$ is 40 degrees. If ray $CA$ is rotated 480 degrees about point $C$ in a clockwise direction, what will be the positive measure of the new acute angle $ACB$, in degrees?

[asy]
draw((0,0)--dir(40),linewidth(1),Arrow);
draw((0,0)--dir(0),linewidth(1),Arrow);
dot(.8dir(40));
dot(.8dir(0));
dot((0,0));
label("A",.8dir(40),NW);
label("B",.8dir(0),S);
label("C",(0,0),W);
[/asy]
Explanation: A rotation by 480 degrees is the same as one by $480 - 360 = 120$.  The first $40$ degrees of this rotation will be used to decrease the angle $ACB$ to $0$ degrees, leaving $80$ unused degrees, so our answer is $\boxed{80}$ degrees.